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            <h1 class="post-title">「线性代数_01」矩阵及其应用</h1>
            
                <div class="post-meta">
                    

                    
                        <span class="post-time">
                        时间： <a href="#">二月 28, 2017&nbsp;&nbsp;14:19:29</a>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
                        </span>
                    
                    
                        <span class="post-category">
                    分类：
                            
                                <a href="/categories/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0/">线性代数</a>
                            
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            <h1 id="矩阵的概念"><a href="#矩阵的概念" class="headerlink" title="矩阵的概念"></a>矩阵的概念</h1><h2 id="矩阵的概念-1"><a href="#矩阵的概念-1" class="headerlink" title="矩阵的概念"></a>矩阵的概念</h2><blockquote>
<p><strong>定义：</strong></p>
<p>由$ m \times n $个数$ a_{ij} ( i=1,2,…,m;j=1,2,…,n ) $排成一个$ m $行$ n $列的矩形数表<br>$$<br>\begin{pmatrix}<br>a_{11} &amp; a_{12} &amp; \dots &amp; a_{1n} \\<br>a_{21} &amp; a_{22} &amp; \dots &amp; a_{2n} \\<br>\vdots &amp; \vdots &amp; &amp; \vdots \\<br>a_{m1} &amp; a_{m2} &amp; \dots &amp; a_{mn}<br>\end{pmatrix}<br>$$</p>
<p>或</p>
<p>$$<br>\begin{bmatrix}<br>a_{11} &amp; a_{12} &amp; \dots &amp; a_{1n} \\<br>a_{21} &amp; a_{22} &amp; \dots &amp; a_{2n} \\<br>\vdots &amp; \vdots &amp; &amp; \vdots \\<br>a_{m1} &amp; a_{m2} &amp; \dots &amp; a_{mn}<br>\end{bmatrix}<br>$$</p>
<p>称为$ m \times n $矩阵或$ m $行$ n $列矩阵，简称<strong>矩阵</strong>.横排称为矩阵的<strong>行</strong>，纵排称为矩阵的<strong>列</strong>，$ a_{ij} ( i=1,2,…,m;j=1,2,…,n ) $称为矩阵的第$ i $行第$ j $列<strong>元</strong>或$ (i,j) $<strong>元</strong>.$ m \times n $矩阵通常用大写字母如$ \pmb{A},\pmb{B},… $表示，有时也记作$ \pmb{A} = ( a_{ij} )_{m \times n} $.</p>
</blockquote>
<hr>
<blockquote>
<p><strong>注意</strong>：</p>
<ul>
<li><strong>实矩阵</strong>：元都是实数的矩阵.</li>
<li><strong>复矩阵</strong>：元都是复数的矩阵.</li>
<li><strong>零矩阵</strong>：元都是$ 0 $的矩阵，记为$ \pmb{O} $.</li>
<li><strong>列矩阵</strong>：只有一列的矩阵，也称为列向量.</li>
</ul>
<p>$$<br>\pmb{A} =<br>\begin{pmatrix}<br>a_1 \\<br>a_2 \\<br>\vdots \\<br>a_m<br>\end{pmatrix}<br>$$</p>
<ul>
<li><strong>行矩阵</strong>：只有一行的矩阵，也称为行向量.</li>
</ul>
<p>$$<br>\pmb{B} =<br>\begin{pmatrix}<br>b_1 &amp; b_2 &amp; \dots &amp;    b_n<br>\end{pmatrix}<br>$$</p>
<ul>
<li><p>行向量和列向量也可以用小写字母$ \pmb{a}, \pmb{b},…; \pmb{\alpha} , \pmb{\beta} , … $表示.</p>
</li>
<li><p>若$ m=n $，即矩阵的行数与列数相同时，称矩阵为<strong>$ n $阶矩阵</strong>或<strong>$ n $阶方阵</strong>.在$ n $阶矩阵<strong>$ A $</strong>中，从左上角到右下角的对角线称为$ \pmb{A} $的<strong>主对角线</strong>，主对角线上的元$ a_{ii} $称为$ n $阶矩阵$ \pmb{A} $的主对角线元；从右上角到左下角的对角线称为$ \pmb{A} $的<strong>次对角线</strong>.</p>
</li>
</ul>
</blockquote>
<h2 id="几种特殊矩阵"><a href="#几种特殊矩阵" class="headerlink" title="几种特殊矩阵"></a>几种特殊矩阵</h2><ul>
<li><p><strong>上三角（形）矩阵</strong>：主对角线以下的元全为$ 0 $的$ n $阶方阵.<br>$$<br>\pmb{A} =<br>\begin{pmatrix}<br>a_{11} &amp; a_{12} &amp; \dots &amp; a_{1n} \\<br>0 &amp; a_{22} &amp; \dots &amp; a_{2n} \\<br>\vdots &amp; \vdots &amp; &amp; \vdots \\<br>0 &amp; 0 &amp; \dots &amp; a_{nn}<br>\end{pmatrix}<br>$$</p>
</li>
<li><p><strong>下三角（形）矩阵</strong>：主对角线以上的元全为$ 0 $的$ n $阶方阵.</p>
</li>
</ul>
<p>$$<br>\pmb{B} =<br>\begin{pmatrix}<br>b_{11} &amp; 0 &amp; \dots &amp; 0 \\<br>b_{21} &amp; b_{22} &amp; \dots &amp; 0 \\<br>\vdots &amp; \vdots &amp; &amp; \vdots \\<br>b_{n1} &amp; b_{n2} &amp; \dots &amp; b_{nn}<br>\end{pmatrix}<br>$$</p>
<ul>
<li><p><strong>三角形矩阵</strong>：上三角形矩阵和下三角形矩阵统称为三角形矩阵.</p>
</li>
<li><p><strong>对角矩阵（diagonal matrix）</strong>：如果$ n $阶方阵的主对角线以外的元全为$ 0 $，则称为对角矩阵，记作$ \pmb{\Lambda} $或$ diag(a_{11} , a_{22} , \dots , a_{nn}) $.<br>$$<br>\pmb{\Lambda} =<br>\begin{pmatrix}<br>a_{11} &amp; &amp; &amp; \\<br>&amp; a_{22} &amp; &amp; \\<br>&amp;  &amp; \ddots &amp; \\<br>&amp;  &amp;  &amp; a_{nn}<br>\end{pmatrix}<br>$$</p>
</li>
</ul>
<blockquote>
<p>对角矩阵对角线上可以有零元.</p>
</blockquote>
<ul>
<li><p><strong>单位矩阵</strong>：主对角线上的元全为$ 1 $的$ n $阶对角矩阵称为$ n $阶单位矩阵，记作$ E_n $，$ I_n $或$ E $，$ I $.<br>$$<br>E = I =<br>\begin{pmatrix}<br>1 &amp; &amp; &amp; \\<br>&amp; 1 &amp; &amp; \\<br>&amp;  &amp; \ddots &amp; \\<br>&amp;  &amp;  &amp; 1<br>\end{pmatrix}<br>$$</p>
</li>
<li><p><strong>同型矩阵</strong>：两个行数与列数相等的矩阵.</p>
</li>
<li><p><strong>相等矩阵</strong>：设 $ \pmb{A}=(a_{ij})_{m\times n} $与$ \pmb{B}= (a_{ij})_{m\times n} $是同型矩阵且对应元相等，则称$ \pmb{A} $与$ \pmb{B} $相等，记作$ \pmb{A} = \pmb{B} $.即</p>
</li>
</ul>
<p>$$<br>\pmb{A} = \pmb{B} \Leftrightarrow a_{ij} = b_{ij},i=1,2,…,m;j=1,2,…,n.<br>$$</p>
<ul>
<li><strong>系数矩阵</strong>：在实际问题中，常常会遇到一些变量要用另外一些变量线性表示.设一组变量$ y_1 , y_2 ,…, y_m $用另一组变量$ x_1 , x_2 ,…, x_n $表示为<br>$$<br>\left \lbrace<br>\begin{aligned}<br>y_1 &amp; = a_{11}x_1+a_{12}x_2+…+a_{1n}x_n \\<br>y_2 &amp; = a_{21}x_1+a_{22}x_2+…+a_{2n}x_n \\<br>&amp; \dots \dots \dots \dots \\<br>y_m &amp; = a_{m1}x_1+a_{m2}x_2+…+a_{mn}x_n<br>\end{aligned}<br>\right.<br>$$</li>
</ul>
<p>称此关系式为从变量$ x_1 , x_2 ,…, x_n $到变量$ y_1 , y_2 ,…, y_m $的线性变换.</p>
<p>这个线性变换中的系数组成的矩阵$ {A} $称为此线性变换的系数矩阵.</p>
<p>$$<br>\pmb{A} =<br>\begin{pmatrix}<br>a_{11} &amp; a_{12} &amp; \dots &amp; a_{1n} \\<br>a_{21} &amp; a_{22} &amp; \dots &amp; a_{2n} \\<br>\vdots &amp; \vdots &amp; &amp; \vdots \\<br>a_{m1} &amp; a_{m1} &amp; \dots &amp; a_{mn}<br>\end{pmatrix}<br>$$</p>
<h1 id="矩阵的运算"><a href="#矩阵的运算" class="headerlink" title="矩阵的运算"></a>矩阵的运算</h1><h2 id="矩阵的加法与数乘"><a href="#矩阵的加法与数乘" class="headerlink" title="矩阵的加法与数乘"></a>矩阵的加法与数乘</h2><blockquote>
<p><strong>定义</strong>：</p>
<p>设有两个同型的$ m\times n $矩阵$ \pmb{A}=(a_{ij}),\pmb{B}=(b_{ij}) $. 矩阵$ \pmb{A} $与$ \pmb{B} $的和记作$ \pmb{A}+\pmb{B} $，规定为</p>
<p>$$<br>\pmb{A}+\pmb{B} =<br>\begin{pmatrix}<br>a_{11}+b_{11} &amp; a_{12}+b_{12} &amp; \dots &amp; a_{1n}+b_{1n} \\<br>a_{21}+b_{21} &amp; a_{22}+b_{22} &amp; \dots &amp; a_{2n}+b_{2n} \\<br>\vdots &amp; \vdots &amp; &amp; \vdots \\<br>a_{m1}+b_{m1} &amp; a_{m1}+b_{m1} &amp; \dots &amp; a_{mn}+b_{mn}<br>\end{pmatrix}<br>$$</p>
</blockquote>
<hr>
<blockquote>
<p><strong>定义</strong>：</p>
<p>数$ k $与矩阵$ \pmb{A} $的乘积，简称数乘，记作$ k\pmb{A} $或$ \pmb{A}k $,规定为</p>
<p>$$<br>k\pmb{A} = \pmb{A}k =<br>\begin{pmatrix}<br>ka_{11} &amp; ka_{12} &amp; \dots &amp; ka_{1n} \\<br>ka_{21} &amp; ka_{22} &amp; \dots &amp; ka_{2n} \\<br>\vdots &amp; \vdots &amp; &amp; \vdots \\<br>ka_{m1} &amp; ka_{m1} &amp; \dots &amp; ka_{mn}<br>\end{pmatrix}<br>$$</p>
</blockquote>
<hr>
<blockquote>
<p><strong>注意</strong>：</p>
<ul>
<li>矩阵的加法与数乘统称为<strong>矩阵的线性运算</strong>.</li>
<li><strong>负矩阵</strong>：对于矩阵$ \pmb{A}= (a_{ij}) $, 称矩阵$ (-a_{ij}) $为$ \pmb{A} $的负矩阵，记作$ -\pmb{A} $.</li>
<li><strong>矩阵的减法</strong>：由负矩阵可以定义矩阵$ {A} $与$ {B} $的减法为$ \pmb{A}-\pmb{B} = \pmb{A}+(-\pmb{B}) $，即两个同型矩阵相减，归结为它们的对应元相减.</li>
<li>$ \pmb{A} = \pmb{B} \Leftrightarrow \pmb{A}-\pmb{B}=\pmb{O} $.</li>
</ul>
</blockquote>
<hr>
<blockquote>
<p><strong>性质</strong>：</p>
<p>矩阵的线性运算满足下列运算规律（设$ \pmb{A} , \pmb{B} , \pmb{C} $都是$ m\times n $矩阵，$ k $ 与$ l $ 为数）：</p>
<ol>
<li>$ \pmb{A}+\pmb{B}=\pmb{B}+\pmb{A} $.</li>
<li>$ (\pmb{A}+\pmb{B})+\pmb{C} = \pmb{A}+(\pmb{B}+\pmb{C}) $.</li>
<li>$ \pmb{A}+\pmb{O}=\pmb{A} $.</li>
<li>$ \pmb{A}+(-\pmb{A})=\pmb{O} $.</li>
<li>$ 1 \cdot \pmb{A} = \pmb{A} $.</li>
<li>$ (kl)\pmb{A}=k(l\pmb{A}) $.</li>
<li>$ (k+l)\pmb{A}=k\pmb{A}+l\pmb{A} $.</li>
<li>$ k(\pmb{A}+\pmb{B})=k\pmb{A}+k\pmb{B} $.</li>
</ol>
</blockquote>
<h2 id="矩阵的乘法"><a href="#矩阵的乘法" class="headerlink" title="矩阵的乘法"></a>矩阵的乘法</h2><blockquote>
<p><strong>定义</strong>：</p>
<p>设$ \pmb{A}=(a_{ij}) $是一个$ m\times s $矩阵，$ \pmb{B}=(b_{ij}) $是一个$ s\times n $矩阵，规定矩阵 $ \pmb{A} $与矩阵$ \pmb{B} $的乘积是$ m\times n $矩阵$ \pmb{C}=(c_{ij}) $，记为$ \pmb{C}=\pmb{AB} $,其中</p>
<p>$$<br>c_{ij} = a_{i1}b_{1j}+a_{i2}b_{2j}+…+a_{is}b_{sj}=\sum_{k=1}^{s} a_{ik}b_{kj}.(i=1,2,…,m;j=1,2,…,n)<br>$$</p>
</blockquote>
<hr>
<blockquote>
<p><strong>注意</strong>：</p>
<ul>
<li>矩阵乘法不满足交换律，即在一般条件下，$ \pmb{AB} \neq \pmb{BA} $.        </li>
<li>两个非零矩阵之积可能是零矩阵.</li>
<li>若$ \pmb{A}\neq\pmb{O} $，$\pmb{AB}=\pmb{AC} $不能推出$ \pmb{B}=\pmb{C} $.</li>
</ul>
</blockquote>
<hr>
<blockquote>
<p><strong>性质</strong>：</p>
<p>矩阵的乘法满足下面的运算规律：</p>
<ul>
<li>$ (\pmb{AB})\pmb{C}=\pmb{A}(\pmb{BC}) $.</li>
<li>$ \pmb{A}(\pmb{B}+\pmb{C})=\pmb{AB}+\pmb{AC} , (\pmb{B}+\pmb{C})\pmb{A}=\pmb{BA}+\pmb{CA} $.</li>
<li>$ \lambda(\pmb{AB})=(\lambda\pmb{A})\pmb{B}=\pmb{A}(\lambda\pmb{B}) $，其中$ \lambda $是数.</li>
<li>$  \pmb{E}_m\pmb{A}_{m\times n} = \pmb{A}_{m\times n}\pmb{E}_n = \pmb{A}_{m\times n} $.</li>
</ul>
</blockquote>
<h2 id="方阵的幂与多项式"><a href="#方阵的幂与多项式" class="headerlink" title="方阵的幂与多项式"></a>方阵的幂与多项式</h2><p>设$ \pmb{A} $为$ n $阶方阵，$ k $为正整数，$ k $个$ \pmb{A} $的连乘积称为$ \pmb{A} $的$ k $次幂，记作$ \pmb{A}^k $.即</p>
<p>$$<br>\pmb{A}^k=\overbrace{\pmb{A}\pmb{A} … \pmb{A}}^{k}<br>$$</p>
<blockquote>
<p><strong>注意</strong>：$ \pmb{A}^0 = \pmb{E} $.</p>
</blockquote>
<hr>
<blockquote>
<p><strong>性质</strong>：</p>
<p>方阵的幂满足下面运算规律（$ \pmb{A} $为方阵，$ k,l $为非负整数）：</p>
<ul>
<li>$ \pmb{A}^k\pmb{A}^l=\pmb{A}^{k+l} $.</li>
<li>$ (\pmb{A}^k)^l=\pmb{A}^{kl} $.</li>
</ul>
</blockquote>
<hr>
<blockquote>
<p><strong>注意</strong>：由于矩阵乘法不满足交换律，一般地，$ (\pmb{AB})^k\neq \pmb{A}^k\pmb{B}^k $.</p>
</blockquote>
<h2 id="矩阵的转置"><a href="#矩阵的转置" class="headerlink" title="矩阵的转置"></a>矩阵的转置</h2><h2 id="共轭矩阵"><a href="#共轭矩阵" class="headerlink" title="共轭矩阵"></a>共轭矩阵</h2><h2 id="可逆矩阵"><a href="#可逆矩阵" class="headerlink" title="可逆矩阵"></a>可逆矩阵</h2><h1 id="分块矩阵"><a href="#分块矩阵" class="headerlink" title="分块矩阵"></a>分块矩阵</h1><h1 id="矩阵的初等变换"><a href="#矩阵的初等变换" class="headerlink" title="矩阵的初等变换"></a>矩阵的初等变换</h1><h2 id="高斯消元法"><a href="#高斯消元法" class="headerlink" title="高斯消元法"></a>高斯消元法</h2><h2 id="初等变换"><a href="#初等变换" class="headerlink" title="初等变换"></a>初等变换</h2><h1 id="初等矩阵"><a href="#初等矩阵" class="headerlink" title="初等矩阵"></a>初等矩阵</h1>
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